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4. Process Modeling
4.6. Case Studies in Process Modeling
4.6.3. Ultrasonic Reference Block Study

4.6.3.6.

Work This Example Yourself

View Dataplot Macro for this Case Study This page allows you to repeat the analysis outlined in the case study description on the previous page using Dataplot, if you have downloaded and installed it. Output from each analysis step below will be displayed in one or more of the Dataplot windows. The four main windows are the Output window, the Graphics window, the Command History window and the Data Sheet window. Across the top of the main windows there are menus for executing Dataplot commands. Across the bottom is a command entry window where commands can be typed in.
Data Analysis Steps Results and Conclusions

Click on the links below to start Dataplot and run this case study yourself. Each step may use results from previous steps, so please be patient. Wait until the software verifies that the current step is complete before clicking on the next step.


The links in this column will connect you with more detailed information about each analysis step from the case study description.

1. Get set up and started.
   1. Read in the data.




                              
 1. You have read 2 columns of numbers 
    into Dataplot, variables the
    ultrasonic response and metal
    distance
2. Plot data, pre-fit for starting values, and
   fit nonlinear model.
   1. Plot the ultrasonic response versus
      metal distance.



   2. Run PREFIT to generate good
      starting values.


   3. Nonlinear fit of the ultrasonic response
      versus metal distance.  Plot predicted
      values and overlay the data.
   4. Generate a 6-plot for model
      validation.

   5. Plot the residuals against
      the predictor variable.


                              
 1. Initial plot indicates that a
    nonlinear model is required.
    Theory dictates an exponential
    over linear for the initial model.

 2. Pre-fit indicated starting
    values of 0.1 for all 3
    parameters.

 3. The nonlinear fit was carried out.
    Initial fit looks pretty good.

 4. The 6-plot shows that the model
    assumptions are satisfied except for
    the non-homogeneous variances.
 5. The detailed residual plot shows
    the non-homogeneous variances
    more clearly.
3. Improve the fit with transformations.
   1. Plot several common transformations
      of the dependent variable (ultrasonic
      response).

   2. Plot several common transformations
      of the predictor variable (metal).


   3. Nonlinear fit of transformed data.
      Plot predicted values with the
      data.


   4. Generate a 6-plot for model
      validation.



   5. Plot the residuals against
      the predictor variable.


 1. The plots indicate that a square
    root transformation on the dependent
    variable (ultrasonic response) is a
    good candidate model.
 2. The plots indicate that no
    transformation on the predictor
    variable (metal distance) is
    a good candidate model.
 3. Carry out the fit on the transformed
    data.  The plot of the predicted
    values overlaid with the data 
    indicates a good fit.

 4. The 6-plot suggests that the model
    assumptions, specifically homogeneous
    variances for the errors, are
    satisfied.

 5. The detailed residual plot shows
    more clearly that the homogeneous
    variances assumption is now
    satisfied.
4. Improve the fit using weighting.
   1. Fit function to determine appropriate
      weight function.  Determine value for
      the exponent in the power model.

   2. Plot residuals from fit to determine
      appropriate weight function.

   3. Weighted linear fit of field versus
      lab.  Plot predicted values with
      the data.

   4. Generate a 6-plot for model
      validation.

   5. Plot the residuals against
      the predictor variable.


 1. The fit to determine an appropriate
    weight function indicates that a
    value for the exponent in the range
    -1.0 to -1.1 should be reasonable.
 2. The residuals from this fit 
    indicate no major problems.

 3. The weighted fit was carried out.
    The plot of the predicted values
    overlaid with the data suggests
    that the variances arehomogeneous.
 4. The 6-plot shows that the model
    assumptions are satisfied.

 5. The detailed residual plot suggests
    the homogeneous variances for the
    errors more clearly.
5. Compare the fits.
   1. Plot predicted values from each
      of the three models with the 
      data.



 1. The transformed and weighted fits
    generate only slightly different
    predicted values, but the model
    assumptions are not violated.

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